Explanation of 1 mod 3

Question

So i've been looking into modulo recently. I'm trying to improve my math skills, which are not the best if i'm honest. But something i am trying to improve. I understand how this works i think. I am also quite competent with long division. However something is bugging me and i can't seem to find an answer for it online.

I know that 7 % 5 = 2 (5 goes into 7 once, with a remainder of 2).

What i don't understand is this;

1 % 3 = 1

How can this be, 3 goes into 1, 0 times, with a remainder of 3? Surely the answer to 1 % 3 = 3?

Can anyone explain this in its most simplest terms please?

Am i correct in thinking that if the dividend (1) is less than the devisor (3) which we know will equal 0 remainder x, it just uses the dividend as the result?

Thanks for your help.

Answer 1

The remainder in 1%3 refers to what remains of 1 (not 3) after you divide by 3. As you have already said, 3 goes into 1 zero times. So -- when you remove 0 multiples of 3 from 1, all of 1 remains. Thus 1 % 3 = 1.

Answer 2

The result of a modulo operation n % m is just that number r for which q * m + r = n (q may be anything). The only requirement we have is that 0 <= r < m.

So for instance:

7 % 5 --> 1 * 5 + 2 == 7 --> r = 2
1 % 3 --> 0 * 3 + 1 == 1 --> r = 1